From Backward Approximations to Lagrange Polynomials in Discrete Advection–Reaction Operators

  • Francisco J. Solis
  • Ignacio Barradas
  • Daniel Juarez
Original Research


In this work we introduce a family of operators called discrete advection–reaction operators. These operators are important on their own right and can be used to efficiently analyze the asymptotic behavior of a finite differences discretization of variable coefficient advection–reaction–diffusion partial differential equations. They consists of linear bidimensional discrete dynamical systems defined in the space of real sequences. We calculate explicitly their asymptotic evolution by means of a matrix representation. Finally, we include the special case of matrices with different eigenvalues to show the connection between the operators evolution and interpolation theory.


Backward approximation Advection–reaction operators Infinite matrix Lagrange polynomials 


  1. 1.
    Asheghi, R.: Bifurcations and dynamics of a discrete predator-prey system. J. Biol. Dyn. 8(1), 161–186 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bischi, G.I., Naimzada A.K.: A Kaleckian macromodel with memory. In: Cristini, A., Fazzari, S., Greenberg, L., Leoni R. (eds) Cycles, Growth and the Great Recession, Routledge, United Kingdom, pp 103–116 (2015)Google Scholar
  3. 3.
    Burden, R.L.: Numerical analysis. Beooks/Cole, Australia (2001)Google Scholar
  4. 4.
    Cortés, J.C., Navarro-Quiles, A., Romero, J.V., Roselló, M.D., Sohaly, M.A.: Solving the random Cauchy one-dimensional advection-diffusion equation: numerical analysis and computing. J. Comput. Appl. Math. 330, 920–936 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cortés, J.C., Jódar, L., Villafuerte, L., Villanueva, R.J.: Computing mean square approximations of random diffusion models with source term. Math. Comput. Simul. 76(1–3), 44–48 (2007). 12MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chong, T.H.: A variable mesh finite difference method for solving a class of parabolic differential equations in one space variable. SIAM J. Numer. Anal. 15, 835–857 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    El-Sayed, A.M.A., Elsadany, A.A., Awad, A.M.: Chaotic dynamics and synchronization of cournot duopoly game with a logarithmic demand function. Appl. Math. 9(6), 3083–3094 (2015)Google Scholar
  8. 8.
    Forsythe, G.E., Wasow, W.R.: Finite difference methods for partial differential equations. Wiley, New York (1964)zbMATHGoogle Scholar
  9. 9.
    Gladwell, I., Wait, R.: A survey of numerical methods for partial differential equations. Oxford University Press, New York (1979)zbMATHGoogle Scholar
  10. 10.
    Jerez, S., Gonzalez, L.M., Solis, F.J.: A regular perturbation analytical-numerical method for the evolution of precancerous lesions caused by the human papillomavirus. Numer. Methods Partial Differ. Eq. 31(3), 847–855 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Khodabin, M., Maleknejad, K., Rostami, M., Nouri, M.: Numerical solution of stochastic differential equations by second order Runge-Kutta methods. Math. Comput. Model. 53, 1910–1920 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li, B., He, Z.: Bifurcations and chaos in a two-dimensional discrete HindmarshRose model. Nonlinear Dyn. 76(1), 697–715 (2014)CrossRefzbMATHGoogle Scholar
  13. 13.
    Maire, S., Nguyen, G.: Stochastic finite differences for elliptic diffusion equations in stratified domains. Math. Comput. Simul. 121, 146–165 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pareek, N.K., Patidar, V., Sud, K.K.: Discrete chaotic cryptography using external key. Phys. Lett. A 309, 75–82 (2003). 12, 17MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Reynolds Jr., A.C.: Convergent finite difference schemes for non-linear parabolic equations. SIAM J. Numer. Anal. 9, 523–533 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Schulman, L.S., A Seiden, P.E.: Statistical mechanics of a dynamical system based on Conways game of Life. J. Stat. Phys. 19(3), 293–314 (1978)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Solis, F., Gonzalez, L.M.: A model for HVP infected cells at different lesion discrete stages. Int. J. Complex Syst. Sci. 2(1), 7–10 (2012)Google Scholar
  18. 18.
    Solis, F., Gonzalez, L.M.: Modeling the effects of Human Papilloma Virus in cervical cells. Int. J. Comput. Math. 91, 1–9 (2013)Google Scholar
  19. 19.
    Solis, F., Barradas, I.: Discrete multiple delay advection-reaction operators. J. Comput. Appl. Math. 291, 441–448 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Solis, F.: Dynamical properties of families of discrete delay advection-reaction operators. J. Differ. Equ. Appl. 19(8), 1218–1226 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Solis, F.: Families of discrete advection-reaction operators via divided differences. Appl. Math. Lett. 25, 775–778 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Vasilyev, R.V., et al.: Solution of the stokes equation in three-dimensional geometry by the finite-difference method. Math. Models Comput. Simul. 8(1), 63–72 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Weiser, A., Wheeler, M.F.: On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25(2), 351–375 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2018

Authors and Affiliations

  1. 1.CIMATGuanajuato GtoMexico

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